A wheel with 8 metallic spokes each 50 cm long is rotated with a speed of 120 rev/min in a plane normal to the horizontal component of the earth's magnetic field. The earth's magnetic field at the place is 0.4 G and the angle of dip is \( 60^\circ \). The induced emf between the axle and the rim of the wheel is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 3.14\times10^{-5}\,\mathrm{V} \)
Each spoke is a conducting rod rotating about one end in a magnetic field perpendicular to its plane of rotation. Its motional emf is
\( \displaystyle e = \frac{1}{2} B \omega l^{2} \)
The wheel turns in a plane normal to the horizontal component of the earth's field, so only that component is cut by the spokes. With the angle of dip \( \delta \):
\( \displaystyle B_H = B\cos\delta \)
\( B = 0.4\,\mathrm{G} = 0.4\times10^{-4}\,\mathrm{T} \)
\( B_H = 0.4\times10^{-4} \times \cos 60^\circ = 2\times10^{-5}\,\mathrm{T} \)
\( \displaystyle \omega = 2\pi \times \frac{120}{60}\,\mathrm{rad\,s^{-1}} = 4\pi\,\mathrm{rad\,s^{-1}} \)
\( l = 50\,\mathrm{cm} = 0.5\,\mathrm{m} \)
\( \displaystyle e = \frac{1}{2} \times 2\times10^{-5}\,\mathrm{T} \times 4\pi\,\mathrm{rad\,s^{-1}} \times (0.5\,\mathrm{m})^{2} \)
\( \displaystyle e = \pi\times10^{-5}\,\mathrm{V} \approx 3.14\times10^{-5}\,\mathrm{V} \)
All 8 spokes run between the same axle and the same rim, so they are in parallel. The emf between axle and rim is that of one spoke; the number of spokes does not change it.